This is part of an algebraic topology problem list, maintained by Mark Hovey.

- The biggest problem, in my opinion, is to come up with a specific
vision of where homotopy theory should go, analogous to the Weil
conjectures in algebraic geometry or the Ravenel conjectures in our
field in the late 70s. You can't win the Fields Medal without a Fields
Medal-winning problem; Deligne would not be DELIGNE without the Weil
conjectures and Mike Hopkins would not be MIKE HOPKINS without the
Ravenel conjectures. We can't all be Deligne or Mike, but making the
conjectures requires different talents than proving them, and more of us
might have a chance. This was actually my motivation for making this
list; to provide a forum for conjectures so that we might collectively
be able to form a program analogous to the Weil conjectures. This would
make a huge difference to our field, I think. Of course, they have to
be somewhat accessible conjectures, which the problems below may not be!

- The generating hypothesis, which asserts that the stable homotopy
functor is faithful on the category of finite spectra. That is, if f is
a map of finite spectra such that pi_* f is 0, then f is nullhomotopic.
An unbelievable consequence of this is that the stable homotopy functor
is full as well. This conjecture has withstood serious attempts for
many years, so be careful! The basic reference is P. Freyd, Stable
homotopy, in {\it Proc. Conf. Categorical Algebra (La Jolla, Calif.,
1965)}, 121--172, Springer, New York, 1966; MR {\bf 35} \#2280.
Devinatz and Hopkins have a program to prove the generating hypothesis
when the target is a sphere; see E. S. Devinatz, The generating
hypothesis revisited, in {\it Stable and unstable homotopy (Toronto, ON,
1996)}, 73--92, Amer. Math. Soc., Providence, RI, ; CNO CMP 1 622 339.

- Find some geometric meaning for elliptic cohomology. I believe
this problem may be solvable--we keep learning new things about it. One
thing I will say here; if I am called to referee a paper on elliptic
cohomology that does not deal with the Hopkins viewpoint on elliptic
spectra, I will almost surely reject it. The time is gone when one
could write papers about the Landweber-Ravenel-Stong elliptic cohomology
based on the Jacobi quartic--we now understand that that is only one of
many different elliptic cohomology theories, and all papers on elliptic
cohomology should now accept that and deal with it. The fundamental
reference here is M. J. Hopkins, Topological modular forms, the Witten
genus, and the theorem of the cube, in {\it Proceedings of the
International Congress of Mathematicians, Vol.\ 1, 2 (Z\"urich, 1994)},
554--565, Birkh\"auser, Basel, 1995; MR 97i:11043. But one should also
see Grojnowski's approach to equivariant elliptic
cohomology--unfortunately, this does not seem to be published, but there
is a preprint. Matthew Ando has also thought about this, see M. Ando,
Power operations in elliptic cohomology and representations of loop
groups, Trans. Amer. Math. Soc. ; CNO CMP 1 637 129. I am sure I have
left something out here as well.

- On the same theme, find some way of doing index theory related to
elliptic cohomology. This is not really algebraic topology, but would
have a major impact on our field. I don't know much about this, but
people who might are Ezra Getzler and Richard Melrose on the analysis
side. Richard Melrose has a theory of index theory on manifolds with
corners that might possibly be relevant, and Ezra has worked on index
theory on certain infinite dimensional manifolds, which again might be
relevant. From the algebraic topology side, I know Haynes Miller and
Mike Hopkins have thought about this some.

- The chromatic splitting conjecture, which is considerably more
complicated to state. Basically nothing is known about this, and so
this one may be more accessible. Besides Hopkins and me, Nori Minami
and Ethan Devinatz have both thought about this conjecture, so might be
good resources. See M. Hovey, Bousfield localization functors and
Hopkins' chromatic splitting conjecture, in {\it The \v Cech centennial
(Boston, MA, 1993)}, 225--250, Contemp. Math., 181, Amer. Math. Soc.,
Providence, RI, 1995; MR 96m:55010 .

- The telescope conjecture. This is one is almost certainly wrong,
but we have all been waiting for 10 years now for a correct disproof of
it to appear. Mahowald, Ravenel, and Shick have claimed to have a
disproof for a long time, and I would guess there is at least a 99%
chance that they are right. But even so, here we have a major
conjecture whose disproof is going to be incomprehensible to almost
everybody in the field. Compare this to the nilpotence theorem or the
smashing theorem, whose proofs have been read and understood by many of
us. So there may be some room here to come up with a more conceptual
approach to the disproof; then again, there may not.

- Classify all finite loop spaces. This is the long term project of
Bill Dwyer and Clarence Wilkerson. The theory, I believe, is that the
Lie groups are essentially the only examples. So one constructs Weyl
groups and maximal tori and the like. But certainly at individual
primes there can be other examples, like BD_3 at p=2.

- Say something general about the stable or unstable homotopy groups
of spheres. For example, Ravenel has suggested that the size of the nth
homotopy group of S^k grows polynomially in n, maybe even cubically. I
presume he meant stable homotopy, but one could also ask the question
unstably.

- The Kervaire invariant problem. This one is on the list not for
its intrinsic importance (in my opinion!) but rather because so many,
many people claim to have solved it, but all their proofs are wrong.
The question is whether the element represented by h_i^2 in the Adams
spectral sequence at the prime 2 is a permanent cycle or not. I am not
sure what reference to give here, but Mark Mahowald is the world's
expert on this problem and all related problems.

- Once again, I am not sure whether this problem deserves to be
called major, but it is annoying that the the R. Cohen - Goerss result
proving that h_0 h_i is a permanent cycle in the Adams spectral sequence
for all primes bigger than 3 is wrong. The flaw was found by Minami,
and it appears to be fatal to their proof. So this problem is still
open.

Department of Mathematics

Wesleyan University

mhovey@wesleyan.edu